Clinton County District Attorney Andrew Wylie said Sweat and Matt had several close calls, including hearing the voices of people nearby when they were hiding in a hunting cabin, CBS News correspondent Anna Werner reports.
Sweat also told police that after the two split up he hid in a tree stand used by hunters and saw a law enforcement officer walk right by him, Werner reports.
Sweat is giving more details to investigators about how the pair escaped from the prison.
Wylie confirmed that Sweat told police that he never used power tools, only a hacksaw blade to cut through steel cell block walls and steam pipes.
Sweat admitted he and Matt conducted a practice run of their prison break on June 5, the night before they escaped.
The men poked their heads out of a manhole, but decided it was too close to nearby homes. Instead, they exited the tunnel early the next morning through a slightly more isolated manhole located just a block from the prison walls.
Matt was shot to death last Friday in Malone by a border patrol officer. Sweat was wounded Sunday by a state trooper near the Canadian border.
Top prison officials have been placed on paid administrative leave. A New York state official told CBS News those prison executives include Superintendent Steven Racette, First Deputy Superintendent Donald Quinn and Deputy Superintendent for Security Stephen Brown.
Racette led New York Gov. Andrew Cuomo on a tour of the prison following the inmates' escape.
Cherie Racette, the superintendent's wife, told the Adirondack Daily Enterprise he was given the option of taking a demotion or retiring and chose retirement. She said he and two deputies are being made scapegoats by Cuomo.
Nine security staffers at the prison have also been placed on paid leave.
"All the staff are hardworking, they're dedicated, and they're great people," retired Clinton Corrections Lt. Michael Blaine, who worked at the prison, told Werner.
Blaine called the shakeup a disruption to other prison employees.
"The staff really do appreciate them, and they do listen to them," Blaine told Werner. "This is very knee-jerk, and it's going to be very dangerous for the facility."
The FBI is investigating alleged corruption at the prison and a possible drug ring.
"The drugs are everywhere," Blaine told Werner. "They're introduced by visitors. They try to get them through [the] package room in different ways, and we're pretty good at getting them, but it's quite the flow."
Prosecutors have said tailor shop instructor Joyce Mitchell got close to the men, supplied them with hacksaw blades and other tools, and agreed to be their getaway driver but backed out at the last moment. She has pleaded not guilty.
Guard Gene Palmer was also arrested, telling investigators he gave the convicts such things as tools, art supplies and access to a catwalk electrical box in exchange for paintings by Matt. But he said he never knew of their escape plans.
Sweat's condition was upgraded from serious to fair at the Albany hospital where he was taken after being shot twice by a state trooper.
Matt's estranged son claimed his dead father's body, the son's mother, Vee Harris told CBS News. Nick Harris earlier told CBS Buffalo affiliate WIVB-TV he believes his father's death was wrongful because agents shot him three times in the head.
Sweat, 35, had been serving life without parole in the killing of a sheriff's deputy. Matt, 49, was doing 25 years to life for the kidnapping and hacksaw dismemberment of his former boss.